30. In a GP, if the first term a = 4 and the common ratio r = 3, what is the 4th term?
4 × 33 = 108
4 + 3 × 4
4 × 32
4 + 4 × 3
Show Solution
The 4th term is:
T4 = 4 × 33 = 108.
39. What is the formula for the area of a triangle?
1/2 × base × height
base × height
base × height × length
1/3 × base × height
Show Solution
The formula is defined as:
Area = 1/2 × base × height.
40. What is the perimeter of a rectangle with length 4 and width 6?
20
24
18
12
Show Solution
Perimeter is calculated as:
2 × (length + width) = 2 × (4 + 6) = 20.
41. What is a matrix?
A rectangular array of numbers
A single number
A collection of equations
An operation on numbers
Show Solution
A matrix is defined as:
A rectangular array of numbers arranged in rows and columns.
42. How many rows does a 3x4 matrix have?
3
4
12
7
Show Solution
A 3x4 matrix has:
3 rows and 4 columns.
43. Which of the following is a square matrix?
2x3
3x3
4x2
1x4
Show Solution
A square matrix has equal number of rows and columns:
The matrix 3x3 is a square matrix.
44. What is the result of adding two matrices of the same dimensions?
A matrix
A scalar
A determinant
A vector
Show Solution
The result of adding two matrices of the same dimensions is:
A matrix formed by adding corresponding elements.
45. What type of matrix is symmetric?
A matrix that is equal to its transpose
A matrix with all zero elements
A matrix with only one row
A rectangular matrix
Show Solution
A symmetric matrix is defined as:
A matrix that is equal to its transpose.
46. What is a skew-symmetric matrix?
A matrix where the transpose is equal to the negative of the original matrix
A matrix with all positive elements
A matrix with equal rows and columns
A matrix with zero diagonal elements
Show Solution
A skew-symmetric matrix is defined as:
A matrix where the transpose is equal to the negative of the original matrix.
47. Which of the following operations can be performed on matrices?
Addition and multiplication
Only addition
Only subtraction
Only multiplication
Show Solution
Matrices can undergo various operations such as:
Addition, subtraction, and multiplication.
48. What is the determinant of a 2x2 matrix?
ad - bc
a + b + c + d
a² + b²
a × d
Show Solution
The determinant of a 2x2 matrix is calculated as:
det(A) = ad - bc, where A = {{a, b}, {c, d}}.
49. How do you calculate the inverse of a 2x2 matrix?
1/det(A) × adj(A)
adj(A) + det(A)
A × A
det(A) + 1
Show Solution
The inverse of a 2x2 matrix A is calculated as:
A-1 = 1/det(A) × adj(A).
50. What are minors in a determinant?
Determinants of smaller matrices
The coefficients of the original matrix
The products of the diagonal elements
The sums of the elements
Show Solution
Minors are defined as:
Determinants of the smaller matrices obtained by deleting one row and one column from the original matrix.
51. What is the concept of co-factors in determinants?
Minors multiplied by (-1)^(i+j)
Sum of the matrix elements
Products of the diagonal elements
The determinants of all elements
Show Solution
Co-factors are defined as:
Minors multiplied by (-1) raised to the sum of their row and column indices.
52. What does the expansion of a determinant refer to?
The process of calculating a determinant using minors and co-factors
Finding the inverse of a matrix
Simplifying a matrix
Performing matrix multiplication
Show Solution
The expansion of a determinant refers to:
The process of calculating a determinant using minors and co-factors.
53. Which property states that the determinant changes sign when two rows are swapped?
Row swapping property
Diagonal property
Multiplication property
Addition property
Show Solution
The property that states that the determinant changes sign when two rows are swapped is called:
The row swapping property.
54. Which matrix is the identity matrix for 2x2 matrices?
[[1, 0], [0, 1]]
[[0, 1], [1, 0]]
[[1, 1], [0, 0]]
[[0, 0], [1, 1]]
Show Solution
The identity matrix for 2x2 matrices is defined as:
[[1, 0], [0, 1]].
55. Which operation does NOT change the determinant of a matrix?
Adding a multiple of one row to another
Swapping two rows
Multiplying a row by a scalar
Adding two matrices
Show Solution
The operation that does NOT change the determinant of a matrix is:
Adding a multiple of one row to another.
56. What is the relationship between a matrix and its transpose?
The rows become columns
The matrix remains unchanged
The determinant doubles
The identity matrix is formed
Show Solution
The relationship between a matrix and its transpose is:
The rows of the original matrix become columns in the transpose.
57. Which of the following matrices is singular?
A matrix with a determinant of zero
A matrix with non-zero determinant
A diagonal matrix
An identity matrix
Show Solution
A singular matrix is defined as:
A matrix with a determinant of zero.
58. How many zeros are in a zero matrix?
All elements are zero
Only diagonal elements are zero
None
Only one element is zero
Show Solution
A zero matrix is defined as:
A matrix where all elements are zero.
59. What is the effect of multiplying a matrix by zero?
The resulting matrix is a zero matrix
The original matrix remains unchanged
The determinant becomes one
The original matrix doubles
Show Solution
The effect of multiplying a matrix by zero is:
The resulting matrix is a zero matrix.
60. What does the term "elementary row operation" refer to?
Operations that can be performed on the rows of a matrix to simplify it
Operations that can be performed on the columns of a matrix only
Calculating the determinant of a matrix
Transposing a matrix
Show Solution
Elementary row operations are defined as:
Operations that can be performed on the rows of a matrix to simplify it.
1. What is the derivative of \( f(x) = x^2 \)?
2x
x
x^2
2
Show Solution
The derivative is calculated as:
f'(x) = 2x
2. What is the formula for the derivative of a constant?
0
1
c
Undefined
Show Solution
The derivative of a constant is:
f'(c) = 0
3. What is the derivative of \( f(x) = 3x^3 \)?
9x^2
3x^2
6x^2
x^2
Show Solution
The derivative is:
f'(x) = 9x^2
4. What is the product rule of differentiation?
f'(x)g(x) + f(x)g'(x)
f'(x)g'(x)
f(x)g(x)
f(x) + g(x)
Show Solution
The product rule states:
f'(x)g(x) + f(x)g'(x)
5. What is the derivative of \( f(x) = \sin(x) \)?
cos(x)
sin(x)
-sin(x)
1
Show Solution
The derivative is:
f'(x) = cos(x)
6. What is the derivative of \( f(x) = \cos(x) \)?
-sin(x)
sin(x)
cos(x)
-cos(x)
Show Solution
The derivative is:
f'(x) = -sin(x)
7. What is the derivative of \( f(x) = e^x \)?
e^x
1
x
e
Show Solution
The derivative is:
f'(x) = e^x
8. What is the derivative of \( f(x) = \ln(x) \)?
1/x
x
ln(x)
e^x
Show Solution
The derivative is:
f'(x) = 1/x
9. What is the chain rule in differentiation?
f'(g(x)) * g'(x)
f'(x) + g'(x)
f(g(x))
g'(x) / f'(x)
Show Solution
The chain rule states:
f'(g(x)) * g'(x)
10. What is the derivative of \( f(x) = x^3 + 2x^2 - 5 \)?
3x^2 + 4x
x^2 + 2x
3x^3 + 2x^2
2x + 3
Show Solution
The derivative is:
f'(x) = 3x^2 + 4x
11. What is the derivative of \( f(x) = \tan(x) \)?
sec^2(x)
tan(x)
1/cos^2(x)
-sec^2(x)
Show Solution
The derivative is:
f'(x) = sec^2(x)
12. What is the derivative of \( f(x) = x^{-1} \)?
-1/x^2
1/x
-x^{-1}
1/x^2
Show Solution
The derivative is:
f'(x) = -1/x^2
13. What is the derivative of \( f(x) = a^x \) where \( a \) is a constant?
a^x ln(a)
a^x
x^a
ln(a)
Show Solution
The derivative is:
f'(x) = a^x ln(a)
14. What is the derivative of \( f(x) = \sqrt{x} \)?
1/(2√x)
1/x
2√x
√x
Show Solution
The derivative is:
f'(x) = 1/(2√x)
15. What is the derivative of \( f(x) = x^4 \)?
4x^3
x^3
3x^4
x^2
Show Solution
The derivative is:
f'(x) = 4x^3
16. What is the derivative of \( f(x) = 5x^5 \)?
25x^4
5x^4
20x^4
30x^5
Show Solution
The derivative is:
f'(x) = 25x^4
17. What is the derivative of \( f(x) = \log_a(x) \)?
1/(x ln(a))
1/x
ln(a)/x
a^x
Show Solution
The derivative is:
f'(x) = 1/(x ln(a))
18. What is the derivative of \( f(x) = \sin(2x) \)?
2cos(2x)
sin(2x)
cos(2x)
2sin(2x)
Show Solution
The derivative is:
f'(x) = 2cos(2x)
19. What is the derivative of \( f(x) = \tan(3x) \)?
3sec^2(3x)
sec^2(3x)
3tan(3x)
tan(3x)
Show Solution
The derivative is:
f'(x) = 3sec^2(3x)
20. What is the derivative of \( f(x) = x \ln(x) \)?
1 + ln(x)
ln(x)
1/x
x
Show Solution
The derivative is:
f'(x) = 1 + ln(x)
21. What is the derivative of \( f(x) = 2x^2 + 3x + 1 \)?
4x + 3
2x + 3
3x^2 + 2x
2x^2 + 3
Show Solution
The derivative is:
f'(x) = 4x + 3
22. What is the derivative of \( f(x) = 4x^3 - x^2 + 6x - 2 \)?
12x^2 - 2x + 6
12x^2 - x + 6
4x^2 - 2x + 6
12x^2 + 2x + 6
Show Solution
The derivative is:
f'(x) = 12x^2 - 2x + 6
23. What is the derivative of \( f(x) = e^{2x} \)?
2e^{2x}
e^{2x}
2e^{x}
e^{x}
Show Solution
The derivative is:
f'(x) = 2e^{2x}
24. What is the derivative of \( f(x) = \ln(x^2 + 1) \)?
2x/(x^2 + 1)
1/(x^2 + 1)
1/x
2x
Show Solution
The derivative is:
f'(x) = 2x/(x^2 + 1)
25. What is the derivative of \( f(x) = \sin^2(x) \)?
2sin(x)cos(x)
sin^2(x)
cos^2(x)
2sin^2(x)
Show Solution
The derivative is:
f'(x) = 2sin(x)cos(x)
26. What is the derivative of \( f(x) = x^3 - 3x + 2 \)?
3x^2 - 3
3x^2 + 2
2x^3 - 3
3x^2 + 6
Show Solution
The derivative is:
f'(x) = 3x^2 - 3
27. What is the derivative of \( f(x) = x^{1/2} + x^{1/3} \)?
1/(2√x) + 1/(3x^{2/3})
1/(2x) + 1/(3x^{1/3})
1/(3√x) + 1/(2x^{2/3})
1/(3x) + 1/(2x^{1/2})
Show Solution
The derivative is:
f'(x) = 1/(2√x) + 1/(3x^{2/3})
28. What is the derivative of \( f(x) = \tan(x^2) \)?
2xsec^2(x^2)
sec^2(x^2)
2sec^2(x)
tan(x^2)
Show Solution
The derivative is:
f'(x) = 2xsec^2(x^2)
29. What is the derivative of \( f(x) = e^{x^2} \)?
2xe^{x^2}
e^{x^2}
2e^{x}
xe^{x}
Show Solution
The derivative is:
f'(x) = 2xe^{x^2}
30. What is the derivative of \( f(x) = \ln(3x + 4) \)?
3/(3x + 4)
1/(3x + 4)
4/(3x + 4)
1/3
Show Solution
The derivative is:
f'(x) = 3/(3x + 4)
31. What is the derivative of \( f(x) = \cos(4x) \)?
-4sin(4x)
-sin(4x)
4sin(4x)
cos(4x)
Show Solution
The derivative is:
f'(x) = -4sin(4x)
32. What is the derivative of \( f(x) = \sec(x) \)?
sec(x)tan(x)
sec(x)cos(x)
sec^2(x)
1/cos^2(x)
Show Solution
The derivative is:
f'(x) = sec(x)tan(x)
33. What is the derivative of \( f(x) = \tan(3x) \)?
3sec^2(3x)
sec^2(3x)
3tan(3x)
3sin(3x)
Show Solution
The derivative is:
f'(x) = 3sec^2(3x)
34. What is the derivative of \( f(x) = \sqrt{x^2 + 1} \)?
x/(√(x^2 + 1))
1/(2√(x^2 + 1))
1/(√(x^2 + 1))
√(x^2 + 1)
Show Solution
The derivative is:
f'(x) = x/(√(x^2 + 1))
35. What is the derivative of \( f(x) = \sin(2x) \)?
2cos(2x)
sin(2x)
2sin(2x)
cos(2x)
Show Solution
The derivative is:
f'(x) = 2cos(2x)
36. What is the derivative of \( f(x) = 5x^3 - 4x^2 + 7 \)?
15x^2 - 8x
5x^2 - 4x + 7
3x^2 - 4x + 7
15x^2 + 4x + 7
Show Solution
The derivative is:
f'(x) = 15x^2 - 8x
37. What is the derivative of \( f(x) = x^{1/3} + x^{1/4} \)?
1/(3x^{2/3}) + 1/(4x^{3/4})
1/(3x) + 1/(4x^{3/4})
1/(2x^{2/3}) + 1/(3x^{3/4})
1/(4x^{3/4}) + 1/(3x^{1/3})
Show Solution
The derivative is:
f'(x) = 1/(3x^{2/3}) + 1/(4x^{3/4})
38. What is the derivative of \( f(x) = 7\sin(x) + 2\cos(x) \)?
7cos(x) - 2sin(x)
-7cos(x) - 2sin(x)
7cos(x) + 2sin(x)
-7sin(x) - 2cos(x)
Show Solution
The derivative is:
f'(x) = 7cos(x) - 2sin(x)
39. What is the derivative of \( f(x) = x^5 + 3x^3 - x + 4 \)?
5x^4 + 9x^2 - 1
5x^4 + 3x^2 - 1
3x^2 + 5x + 4
5x^4 + 3x^3 + 4
Show Solution
The derivative is:
f'(x) = 5x^4 + 9x^2 - 1
40. What is the derivative of \( f(x) = \frac{1}{x^2} \)?
-\frac{2}{x^3}
Show Solution
The derivative is:
f'(x) = -\frac{2}{x^3}
1. What is the integral of 1/x dx?
ln|x| + C
x + C
1/2x^2 + C
e^x + C
Show Solution
The integral is given by:
∫(1/x) dx = ln|x| + C
2. What is the integral of x^n dx?
nx^(n-1) + C
(1/(n+1))x^(n+1) + C (n ≠ -1)
1/n x^(n-1) + C
ln|x| + C
Show Solution
The integral is calculated as:
∫x^n dx = (1/(n+1))x^(n+1) + C (for n ≠ -1)
3. What is the integral of e^x dx?
e^x + C
x + C
ln|x| + C
e^(x + C)
Show Solution
The integral is given by:
∫e^x dx = e^x + C
4. What is the integral of sin(x) dx?
-cos(x) + C
cos(x) + C
-sin(x) + C
sin(x) + C
Show Solution
The integral is given by:
∫sin(x) dx = -cos(x) + C
5. What is the integral of cos(x) dx?
sin(x) + C
-sin(x) + C
cos(x) + C
-cos(x) + C
Show Solution
The integral is given by:
∫cos(x) dx = sin(x) + C
6. What is the integral of sec^2(x) dx?
tan(x) + C
sec(x) + C
-tan(x) + C
-sec(x) + C
Show Solution
The integral is given by:
∫sec²(x) dx = tan(x) + C
7. What is the integral of csc^2(x) dx?
-cot(x) + C
-cot(x) + C
csc(x) + C
cot(x) + C
Show Solution
The integral is given by:
∫csc²(x) dx = -cot(x) + C
8. What is the integral of sec(x)tan(x) dx?
sec(x) + C
tan(x) + C
-sec(x) + C
-tan(x) + C
Show Solution
The integral is given by:
∫sec(x)tan(x) dx = sec(x) + C
9. What is the integral of 1/(1+x^2) dx?
tan^(-1)(x) + C
cot^(-1)(x) + C
ln|x| + C
1/2ln|1+x^2| + C
Show Solution
The integral is given by:
∫(1/(1+x²)) dx = tan^(-1)(x) + C
10. What is the integral of sinh(x) dx?
cosh(x) + C
sinh(x) + C
-cosh(x) + C
-sinh(x) + C
Show Solution
The integral is given by:
∫sinh(x) dx = cosh(x) + C
11. What is the integral of cosh(x) dx?
sinh(x) + C
cosh(x) + C
-sinh(x) + C
-cosh(x) + C
Show Solution
The integral is given by:
∫cosh(x) dx = sinh(x) + C
12. What is the integral of a^x dx?
(1/ln(a))a^x + C (a > 0, a ≠ 1)
a^x + C
ln|a| + C
x*a^x + C
Show Solution
The integral is given by:
∫(a^x) dx = (1/ln(a))a^x + C (for a > 0, a ≠ 1)
13. What is the integral of 1/sqrt(1-x^2) dx?
sin^(-1)(x) + C
cos^(-1)(x) + C
ln|x| + C
1/2ln|1-x^2| + C
Show Solution
The integral is given by:
∫(1/sqrt(1-x²)) dx = sin^(-1)(x) + C
14. What is the integral of 1/(1+x^2) dx?
tan^(-1)(x) + C
-cot^(-1)(x) + C
ln|1+x^2| + C
1/2ln|x| + C
Show Solution
The integral is given by:
∫(1/(1+x²)) dx = tan^(-1)(x) + C
15. What is the integral of x*sin(x) dx?
-x*cos(x) + sin(x) + C
sin(x) + C
-sin(x) + C
-x*sin(x) + C
Show Solution
Using integration by parts:
∫x*sin(x) dx = -x*cos(x) + sin(x) + C
16. What is the integral of x*cos(x) dx?
x*sin(x) + cos(x) + C
-x*sin(x) + C
sin(x) + C
-x*cos(x) + C
Show Solution
Using integration by parts:
∫x*cos(x) dx = x*sin(x) + cos(x) + C
17. What is the integral of ln(x) dx?
x*ln(x) - x + C
x + C
ln(x) + C
x*ln(x) + C
Show Solution
Using integration by parts:
∫ln(x) dx = x*ln(x) - x + C
18. What is the integral of x^2*e^x dx?
x^2*e^x - 2x*e^x + 2e^x + C
e^x + C
2e^x + C
x^2*e^x + C
Show Solution
Using integration by parts twice:
∫x^2*e^x dx = x^2*e^x - 2x*e^x + 2e^x + C
19. What is the integral of 1/x^2 dx?
-1/x + C
-1/x + C (x ≠ 0)
ln|x| + C
1/x + C
Show Solution
The integral is given by:
∫(1/x²) dx = -1/x + C (for x ≠ 0)
20. What is the integral of e^(-x) dx?
-e^(-x) + C
e^(-x) + C
-e^x + C
e^x + C
Show Solution
The integral is given by:
∫e^(-x) dx = -e^(-x) + C
21. What is the integral of sin(2x) dx?
-1/2 cos(2x) + C
-cos(2x) + C
sin(2x) + C
1/2 sin(2x) + C
Show Solution
The integral is given by:
∫sin(2x) dx = -1/2 cos(2x) + C
22. What is the integral of cos(3x) dx?
1/3 sin(3x) + C
sin(3x) + C
-1/3 sin(3x) + C
-cos(3x) + C
Show Solution
The integral is given by:
∫cos(3x) dx = 1/3 sin(3x) + C
23. What is the integral of sec^2(x) dx?
tan(x) + C
sec(x) + C
ln|sec(x) + tan(x)| + C
sin(x) + C
Show Solution
The integral is given by:
∫sec²(x) dx = tan(x) + C
24. What is the integral of csc^2(x) dx?
-cot(x) + C
csc(x) + C
sec(x) + C
ln|csc(x) + cot(x)| + C
Show Solution
The integral is given by:
∫csc²(x) dx = -cot(x) + C
25. What is the integral of (1 + x^2)^(-1) dx?
tan^(-1)(x) + C
sin^(-1)(x) + C
ln|x| + C
-cos(x) + C
Show Solution
The integral is given by:
∫(1/(1+x²)) dx = tan^(-1)(x) + C
26. What is the integral of sinh(x) dx?
cosh(x) + C
sinh(x) + C
-cosh(x) + C
-sinh(x) + C
Show Solution
The integral is given by:
∫sinh(x) dx = cosh(x) + C
27. What is the integral of cosh(x) dx?
sinh(x) + C
cosh(x) + C
-sinh(x) + C
-cosh(x) + C
Show Solution
The integral is given by:
∫cosh(x) dx = sinh(x) + C
28. What is the integral of a^x dx?
(1/ln(a))a^x + C (a > 0, a ≠ 1)
a^x + C
ln|a| + C
x*a^x + C
Show Solution
The integral is given by:
∫(a^x) dx = (1/ln(a))a^x + C (for a > 0, a ≠ 1)
29. What is the integral of (x^n) dx, where n ≠ -1?
(1/(n+1))x^(n+1) + C
ln|x| + C
x^n + C
n*x^(n-1) + C
Show Solution
The integral is given by:
∫(x^n) dx = (1/(n+1))x^(n+1) + C (for n ≠ -1)
30. What is the integral of (1/x) dx?
ln|x| + C
x + C
1/x + C
-ln|x| + C
Show Solution
The integral is given by:
∫(1/x) dx = ln|x| + C